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What does infinity mean to a physicist

Sigh, allow me to correct some errors.

The only infinitessimal real number is zero.

1/0 is nonsensical, when viewed as arithmetic of real numbers.

Whether or not a mathematical structure has an element called "infinity" is entirely up to the mathematical structure.

I don't know of any (non-contrived) mathematical structures that:
• contain a non-zero infinitessimal number
• that number has an (infinite) reciprocal
• the reciprocal is called "infinity"

All math is, more or less, a "fabrication of the human imagination".

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 Quote by Hurkyl All math is, more or less, a "fabrication of the human imagination".
Yes, I suppose you're right.

Leopold Kronecker has a famous quote saying,

"Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk.
(The dear God has made the integers, all the rest is man's work.)"
-Leopold Kronecker

------------------

Anyway, slightly off topic (or maybe not!) here is fun game to play. I've actually stumped many mathematically inclined people with this riddle (many for well over 10 or 20 minutes).

Proof that 2 = 1
Find the flaw! (game/riddle)

$$(1) \ \ \ \ \ a = x$$

$$(2) \ \ \ \ \ a + a = x + a$$

Simplify left side,

$$(3) \ \ \ \ \ 2a = x+a$$

Subtract 2x from both sides,

$$(4) \ \ \ \ \ 2a - 2x = x + a -2x$$

Factor the left side,

$$(5) \ \ \ \ \ 2(a - x) = x + a -2x$$

Simplify the right side

$$(6) \ \ \ \ \ 2(a - x) = a - x$$

Divide both sides by (a - x)

$$(7) \ \ \ \ \ 2\frac{(a - x)}{(a-x)} = \frac{a - x}{a-x}$$

Simplify

$$(8) \ \ \ \ \ 2 = 1$$

Therefore 2 = 1. QED.

Obviously 2 is not equal to 1, so there must be a flaw in the above logic. Specify the step which has the (first) flaw and the reason it is flawed.

If you already know the answer, or if you figure it out, you might wish to keep it to yourself for awhile (on this thread) lest spoil it for the rest.
 Recognitions: Gold Member Science Advisor Although there are some advanced branches of mathematics where an entity called "infinity" exists, in this post I refer to standard real numbers and complex numbers, which are what most physicists would use. In the standard real number system there is no such number as infinity. Mathematicians do use the words "infinity" or "infinite", but only a shorthand for some other way of expressing it. When being rigorous, you should never write $x = \infty$, but you can write $x \rightarrow \infty$, which has a specific technical meaning. To give an example, you might say that the energy of a particle tends to infinity as the particle's velocity tends to the speed of light ($E \rightarrow \infty$ as $v \rightarrow c$). But this means the same thing as "no matter how much energy you supply to a particle, it will never reach the speed of light". You specify an energy, as large as you like, and the particle can exceed that energy by going fast enough (but still slower than light). Similarly, any sentence containing the word "infinity" or "infinite" can be rephrased in terms of finite quantities. If an equation of physics seems to give you an answer of infinity, that means you've misused the formula and it doesn't apply in that circumstance.
 Thanks DrGreg. Thats about as concise an answer as I could have hoped for.

 I raised this with a colleague, he said from a maths point of view (and presumably physics) that infinite just means immeasurably large but finite.
This is simply not true. In mathematics, something that is finite, no matter how big, cannot be infinite. If he really said that then it reflects poor understanding of basic mathematics principles. Also, if you look at most number sets we use for calculations, such as N (natural numbers), Q (rational numbers) or R (real numbers), none of them include infinity. So if we consider the word 'number' to mean 'real number', then saying 'infinity is the largest possible number' makes no sense, because infinity is not a number.

In the end, mathematics could be considered as just manipulation of symbols. We are free to 'define' infinity in any way we want. Cantor, for example, defined many different types of infinity, some of which were infinitely larger than others. Others have created number systems which include infinity (for example, R plus +/- infinity) but these number systems cause a lot of problems and you need to be very careful to avoid making them self-contradictory. For example, if I naively defined infinity as 1/0, then I could also argue that 2/0 = 2*(1/0) = 2*infinity = infinity, so 2/0 = 1/0 ==> 2*x = 1*x, where x=/=0, so I have proved that 2 equals 1. Even if you manage to create a logical, consistent number system that includes infinity, it would wind up losing many of the nice properties that 'normal' number systems enjoy.

Infinity is, admittedly, a hard concept for beginners in mathematics to understand. In mathematics, we (try to) have a solid set of rules which we follow to reach conclusions, and infinity is no different from anything else in this regard. Infinity does not occupy a special or problematic position in math.
 Thinking back on what my colleague said I probably owe him an apology. his quote was more like "immeasurably large" I think I may have added the "but finite" I come here as a layperson. trained in neither maths nor the other sciences. but I also like to ponder these areas. Thus if something doesn't equate I like to discover why. Usually its just my poor understanding of the subject matter. When physicists discuss matters with people not in their fields its not uncommon for words to mean different things to different people. thus when a word like infinite is used it conjures different images to me (as the layperson) than to say another physicist or mathematician. Say someone says the universe is infinitely large (why do I hear Carl Sagan whenever I think that) then it most probably means a different thing to me than the speaker. To me it means never ending, and not in the way walking around the earth forever is never ending, that is simply repetitious. In the way that traveling at 1 Billion * C in a true straight line (no space time distortion) for trillions of years would result in me being billions of trillions of light years away from the starting point and still the journey has not gone long enough to say its even begun. To me this is clearly not possible. The use of the above description is not meant to provoke a discussion. Infinite acceleration of gravity was another one mentioned in this thread, again clearly impossible when looked at from my understanding of the word infinite. So I sought a better definition of 'Infinite' CC

 Quote by collinsmark Yes, I suppose you're right. Leopold Kronecker has a famous quote saying, "Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk. (The dear God has made the integers, all the rest is man's work.)" -Leopold Kronecker ------------------ Anyway, slightly off topic (or maybe not!) here is fun game to play. I've actually stumped many mathematically inclined people with this riddle (many for well over 10 or 20 minutes). Proof that 2 = 1 Find the flaw! (game/riddle) We start with the postulate that a = x. $$(1) \ \ \ \ \ a = x$$ Add a to both sides. $$(2) \ \ \ \ \ a + a = x + a$$ Simplify left side, $$(3) \ \ \ \ \ 2a = x+a$$ Subtract 2x from both sides, $$(4) \ \ \ \ \ 2a - 2x = x + a -2x$$ Factor the left side, $$(5) \ \ \ \ \ 2(a - x) = x + a -2x$$ Simplify the right side $$(6) \ \ \ \ \ 2(a - x) = a - x$$ Divide both sides by (a - x) $$(7) \ \ \ \ \ 2\frac{(a - x)}{(a-x)} = \frac{a - x}{a-x}$$ Simplify $$(8) \ \ \ \ \ 2 = 1$$ Therefore 2 = 1. QED. Obviously 2 is not equal to 1, so there must be a flaw in the above logic. Specify the step which has the (first) flaw and the reason it is flawed. If you already know the answer, or if you figure it out, you might wish to keep it to yourself for awhile (on this thread) lest spoil it for the rest.
And that's why infinite is non-nonsensical. (**If you can't figure it out, try plugging in numbers)
 My answer was 0=0 not 2=1. But I got it wrong! I also would have stopped at step 4 to proceed would have been wasted effort. When I stepped through the whole thing I got 2=1 So its proof that there is an anomaly in a mathematical supposition. The true answer is 0=0 but if mathematical rules are applied as I know them then 2=1 Are there any similar mathematical absurdities that don't involve that particular rule set? The question this raises to me is, are there any mathematical proofs that rely on this absurdity? If so they should be reworked because even wolfram alpha got it wrong! Please excuse my phrasing year 10 maths is as far as I got. I am not sure what infinity has to do with it. CC

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 Quote by curiouschris even wolfram alpha got it wrong!
I bet it was user error. x/x=1 is a valid identity if x denotes a polynomial. However, if x denotes an indeterminate real number, then x/x=1 is* equivalent to the assertion x is nonzero.

*: In an appropriate syntax where this is not nonsense.
 No not user error at all. Why did wolframalpha assert that 0/0 = 1? Heres my simplistic logic, assuming real numbers. any number divided by itself is 1. This is the rule WA applied but its wrong! Its also the basis for collinsmark's little teaser. The exception is where the number is 0, 0/0 = 0 If you have nothing to start with and divide it by nothing you still have nothing. plain and simple child's logic. Apparently the exception is not taken into account by WA. It also raises what I consider serious concerns about any algebraic formula that doesn't take into account this exception. CC

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 Quote by curiouschris Why did wolframalpha assert that 0/0 = 1?
It looks like an honest-to-goodness software bug -- it looks like wolframalpha is supposed to be using Mathematica to compute a response, but Mathematica's response would be Indeterminate, as you can see with other similar queries like 1*0/0.

The bug may be recent -- I found this page demonstrating it was returning the right answer last May.

 Quote by curiouschris My answer was 0=0 not 2=1. But I got it wrong! I also would have stopped at step 4 to proceed would have been wasted effort. When I stepped through the whole thing I got 2=1 So its proof that there is an anomaly in a mathematical supposition. The true answer is 0=0 but if mathematical rules are applied as I know them then 2=1 CC
uh, no the answer is 0=0. at step four, using law of substitution, x-a=x-x=0. 2*0 is 0, so you can't factor a 2 out of it. the right side of the equation should be 0 as well by law of substitution.
 Thats what I said Billson555. At step 4 the answer is 0=0 and thats why I wouldn't have bothered proceeding. Hurkl, The reason I went to WA was in my head 0/0=0 (not indeterminate) but I wanted to confirm that because anything divide by 0 is well lets say infinite and also Superstring intimated that it would have something to do with that. Remember I am not a mathematician by a long shot. A student of maths may well have accepted 0/0=1 and deduced 1=2 therefore the universe is about to implode. which is exactly what collinsmark was relying on. I was also taught x/x=1 no exceptions. I knew that wasn't acceptable and thats why I said WA was wrong. You assumed I was some sort of dullard and managed to type in something else (normally I do ) Now you say its a software bug. perfectly acceptable explanation and maybe WA should be notified. You also say it should be indeterminate. Theres probably good reason for that but my mind says 0/0=0. What you didn't say was sorry CC

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